Jessie Ware: ‘It’s more soul than you’d think. I want my music to have a timeless quality that isn’t necessarily now.’ Photograph: Katherine Rose for the Observer

One of the many brilliant things to emerge from the legalising of Rinse FM, the London pirate radio station founded in 1994 by DJ Geeneus, has been the artists it has long nurtured. Having pioneered dubstep and grime, Rinse remains a yardstick of London's underground taste which goes some way to explaining how singer Jessie Ware is sitting here today, nursing tea and batting away the compliments.

Ware's trajectory seems logical. First came Rinse FM's break-out vocalist Katy B, spreading dance into the mainstream in 2010, then came Ware, a slightly lo-fi version, albeit with equally big hair. Does she mind comparisons with Katy B? "Nooo!" she cackles. "I've totally got a girl-crush on her."

Like Katy B, 27-year-old Ware started her career providing vocals for artists such as "postdubstepper" SBTRKT (on "Right Thing to Do"), with Sampha (SBTRKT's frequent collaborator) on lo-fi dubstep track, Valentine, and on DJ Joker's "The Vision", where her vocals go a bit Mariah, and wonderfully frenzied. She credits Rinse DJ Oneman with "pioneering" her, yet despite this backing, remained little more than a stalwart vocalist on Boiler Room, YouTube's live stream show, which showcases new music. Then Rinse FM went legal in 2010, dubstep as a genre was revitalised and her music reached a wider audience.

Ware trained classically at primary school, with her tastes drifting into jazz standards by way of Cole Porter and George Gershwin until natural progression "and the 90s" led her to R&B, Lauryn Hill and soul, which is where she's headed with her debut album, out this summer, produced by Dave Okumu of the Invisible. "It's more soul than you'd think. I love Barbra Streisand and Sade who've both had careers in soul and I want my music to have that timeless quality that isn't necessarily now." You get a sense of her neo-soul direction on "Strangest Feeling", her woozy trip-hop debut out last November, and her forthcoming single, "Running", in which Ware abandons the thick beats of youth in favour of laconic soul in the vein of vocalist Diane Charlemagne.

Still, her career in music almost didn't happen. Ware went on to attend Alleyn's school in south London, one of those formidable pro-arts schools (former pupils include Jude Law and Florence Welch). Hers was a strong year – she was friends with Felix White of the Maccabees and Jack Peñate – "but I didn't have the guts to sing, you know? The thought of doing music, like properly? Nah."

Music was always a secondary option, and one which Ware temporarily abandoned for an English degree at Sussex University with the idea of becoming a journalist. She landed a reporter's job at the Jewish Chronicle. "It was wicked. We had smoked salmon bagels on a Friday, all that. Still, I've no idea how I got a job there. Maybe because I'm Jewish, or nosy, or because they respected my dad. Who knows?" Ware's father is Panorama's John Ware, a man who regularly incurs the wrath of Muslim groups following his work tackling Islamic extremism in the UK. "I think deep down I felt it was indulgent to be a singer given what my dad does, which is why it feels nice to be making a success of things."

The turning point for Ware came when Jack Peñate, her "bestest schoolfriend from day one", got her to sing backing vocals at a BBC Maida Vale session before taking her on tour, where Peñate's guitarist introduced her to artists like SBTRKT while driving around America. "A female vocal can lift a song, she says. But she's also aware that recent developments in music industry tastes have been on her side: "Dance music has pushed its way into the mainstream. Which is good for me."

The rest, she says, is down to schmoozing. Ware is naturally affable, gorgeous, too, all yearbook hair, champion eyebrows, dressed in excellent vintage. She's less confident about her abilities, though. "Malleable" is how she describes her voice: "If they want a big vocal, I can do a big vocal. If they want smooth, I can do smooth. I'll do what they want," she smiles.

Modesty apart, compliments have been generous: "SBTRKT said I sounded like Aaliyah. That was fucking nice, I thought. Not bad eh?"